reserve x for set;
reserve i,j for Integer;
reserve n,n1,n2,n3 for Nat;
reserve p for Prime;
reserve a,b,c,d for Element of GF(p);
reserve K for Ring;
reserve a1,a2,a3,a4,a5,a6 for Element of K;
reserve px,py,pz for object;
reserve Px,Py,Pz for Element of GF(p);
reserve P for Element of ProjCo(GF(p));
reserve O for Element of EC_SetProjCo(a,b,p);

theorem Th35:
  for p be Prime, a, b be Element of GF(p),
  P be Element of EC_SetProjCo(a,b,p) holds
  ((P`2_3) |^2)*(P`3_3) - ((P`1_3) |^3 +
    a*(P`1_3)*(P`3_3) |^2 + b*(P`3_3) |^3) = 0.GF(p)
  proof
    let p be Prime, a, b be Element of GF(p),
    P be Element of EC_SetProjCo(a,b,p);
    consider PP be Element of ProjCo(GF(p)) such that
    A1: PP = P & PP in EC_SetProjCo(a,b,p);
    A2: PP`1_3 = P`1_3 & PP`2_3 = P`2_3 & PP`3_3 = P`3_3 by A1,Th32;
    P is_on_curve EC_WEqProjCo(a,b,p) by Th34;
    hence thesis by A2,EC_PF_1:def 8,A1;
  end;
